Applications of noncommutative geometry

Question

This is related to Anweshi's question about theories of noncommutative geometry.

Let's start out by saying that I live, mostly, in a commutative universe. The only noncommutative rings I have much truck with are either supercommutative, almost commutative (filtered, with commutative associated graded), group algebras or matrix algebras, none of which really show many of the true difficulties of noncommutative things.

So, here's my (somewhat pithy) question: what's noncommutative geometry good for?

To be a bit more precise, I have a vague sense that $C^*$ stuff is supposed to work well in quantum mechanics, but I'm somewhat more interested in more algebraic noncommutative geometry. What sorts of problems does it solve that we can't solve without leaving the commutative world? Why should, say, a complex algebraic geometer learn some noncommutative geometry?

Answer 1

Charles,

a couple of reasons why a complex algebraic geometer (certainly someone who is interested in moduli spaces of vector bundles, as your profile tells me) might at least keep an open verdict on the stuff NC-algebraic geometers (NCAGers from now on) are trying to do.

in recent years ,a lot of progress has been made towards understanding moduli spaces of semi-stable representations of 'formally smooth' algebras (think 'smooth in the NC-world). in particular when it comes to their etale local structure and their rationality. for example, there is this book, by someone.

this may not seem terribly relevant to you until you realize that some of the more interesting moduli spaces in algebraic geometry are among those studied. for example, the moduli space of semi-stable rank n bundles of degree 0 over a curve of genus g is the moduli space of representations of a certain dimension vector over a specific formally smooth algebra, as Aidan Schofield showed. he also applied this to rationality results about these spaces.

likewise, the moduli space of semi-stable rank n vectorbundles on the projective plane with Chern classes c1=0 and c2=n is birational to that of semi-simple n-dimensional representations of the free algebra in two variables. the corresponding rationality problem has been studied by NCAG-ers (aka 'ringtheorists' at the time) since the early 70ties (work by S.A. Amitsur, Claudio Procesi and Ed Formanek). by their results, we NCAGers, knew that the method of 'proof' by Maruyama of their stable rationality in the mid 80ties, couldn't possibly work.

it's rather ironic that the best rationality results on these moduli spaces (of bundles over the projective plane) are not due to AGers but to NCAGers : Procesi for n=2, Formanek for n=3 and 4 and Bessenrodt and some guy for n=5 and 7. together with a result by Aidan Schofield these results show that this moduli space is stably rational for all divisors n of 420.

further, what a crepant resolution of a quotient singularity is to you, is to NCAGers the moduli space of certain representations of a nice noncommutative algebra over the singularity.

likewise, when you AGers mumble 'Deligne-Mumford stack', we NCAGers say 'ah! a noncommutative algebra'.

Answer 2

A cool application which I can somehow appreciate is Van den Bergh's proof of dimension $3$ case of the Bondal-Orlov conjecture that two birational smooth Calabi-Yau varieties $X,X'$ have equivalent derived category $D(X) \sim D(X')$. Note that since one can construct the pluricanonical ring from $D(X)$ , this is a generalization of the Batyrev's conjecture that they have the same Hodge numbers. The dimension $3$ case was first proved by Bridgeland, but Van den Bergh's proof uses non-commutative stuff in a very concrete way. Some references can be found in this question I asked.

It goes as follows: by some mimimal model program results, in dimension $3$ any birational $X,X'$ are related by a series of flops $X \to Y \leftarrow X^+$. So we only need to prove $D(X) \sim D(X^+)$. Then one builds a special vector bundle (sum of an exceptional collection of line bundles in this case, some perverse stuff!) on X. Pushforward to $Y$, one gets a coherent sheaf $E$. Let $A= End(E)$. The funny thing is that $D(X) \sim D(A)$. Note that $A$ is non-commutative.

Now, do the same thing for $X+$ one gets $A+$, say. But it is fairly easy to prove that $D(A) \sim D(A^+)$ directly on $Y$, so we are done. If we carry out this on an Atiyah's flop or Reid's pagoda, one can see actually that $A$ and $A+$ are the same. This indicates that the non-commutative route can simplify things.

There seem to be many experts on this site, so surely you will get a lot of much better answers. But this example seems most down to earth for me, and the conjectures come from complex geometry (motivated by physics?) so I hope it helps you as well.

EDIT: here is something to complement Lieven's great answer above: given $A$ one can actually construct back $X$ as a moduli space of certain A-representations (see Section 6 of this). One needs $A$ to have finite global dimension so $X$ can be smooth (the fact that $X$ is smooth is proved via the very algebraic intersection theorem). In dimension $3$, this example explains Lieven's sentence:

further, what a crepant resolution of a quotient singularity is to you, is to NCAGers the moduli space of certain representations of a nice noncommutative algebra over the singularity.

Answer 3

Because there are several approaches to noncommutative algebraic geometry. I will just briefly talk about the motivation of Rosenberg's machine(categorical geometry) and Kontsevich-Rosenberg's machine(functor view point).

The main motivation for Rosenberg's approach is representation theory. Spectrum of abelian category is an important notion in this approach which provided the suitable language to talk about the irreducible representations. For example, say module over noncommutative algebra. So the irreducible representations of this noncommutative algebra is one to one corresponding to the closed points of this spectrum. If we have a noncommutative algebra A, and we have subalgebra B(not any subalgebra,the choice of B is depended on a theorem proved in this framework). What we always do is constructing representations of A from representations of B(which called induction). Using the language of spectrum of abelian category. This induction process can be viewed as the morphism from spectrum of B-mod to Spectrum of A-mod. Well, why is it good? Rosenberg developed a machine dealing with these things. He proved a theorem(I will not mention it at the moment)which allows one to construct all the irreducible representations of A from irreducible representations of B(which is easy to see) in a very functorial way. This theorem is just like an algorithm. Using this method, we can also find some generic representations of B(which correspondence to the generic points of the spectrum).

In fact, he introduced a class of algebra called hyperbolic algebra(some other people called it generalized weyl algebra). Many noncommutative associative algebra in representation theory and mathematical physics are of this kind. For example, n-th weyl algebra, Heisenberg algebra, enveloping algebra of Lie algebra and their quantum analogue. One can used the machine he developed construct the all irreducible representations of these algebras very easily and canonically(I'd like to say, it is somehow like the "automata" way).

I'd like to point out what I talked above is how spectral theory of abelian category comes into representation theory. But this is just a start of this game. More interesting machinery in this framework is one can reduced the representation of some associative algebra(say enveloping algebra of Lie algebra or quantum analogue)to representations of hyperbolic algebra(say, weyl algebra and some others). This story is very interesting because this process can be viewed as find affine cover for noncommutative scheme:

Let me elaborate a bit: It is well known to all representation theorist, we have Beilinson-Bernstein localization for Lie algebra. From the point of view of noncommutative algebraic geometry. category of D-modules on flag variety of Lie algebra is a certain kind of noncommutative projective scheme. We can construct the affine cover(D-module on affine space)for this noncommutative projective scheme which is category of module over Weyl algebra. Now, we have reduced the representations of Lie algebra to the representations of Weyl algebra. Then, using the Gluing machinery of Rosenberg's(application of Barr-Beck's theorem). We can construct irreducible representations of Lie algebra(global)from that of Weyl algebra(local). Additonally,for quantum group, we can apply this framework directly to get representations of quantum group(noncommutative scheme)from reps of algebra of quantum differential operator(analogue of Weyl algebra)(affine cover).There is a theorem which supported the method I mentioned above, called Locality theorem.

Answer 4

As I mentioned in my post on the other question, there are two very fertile applications to Non-commutative Algebraic Geometry in the style of A. Rosenberg;

1. Representation Theory
2. Physics

I provided references in my post over there for both of these, but here are some again

1. Rep Theory: First, Second.
2. Physics: First, Second, Third.

Answer 5

I do not think NCG arose as a way to solve problems in algebraic geometry using new methods. The motivation seems to be to broaden your horizons further.

The answer of Bischof to the question you have cited, gives many contact points with classical topics in algebraic geometry such as deformation theory, invariant theory, moduli spaces, etc..

Also see this article of Lieven le Bryun, in which he speaks of points that can "talk to each other" via common tangent information. In other words, the "Chinese Remainder Theorem" fails for noncommutative rings and so points can be exceedingly close to each other. This is a more interesting way of looking at more general spaces.

I suggest that you look more formally at the "noncommutative torus", which is a very important example in NCG. This space is a quintessential example of the above property of points being very close to each other.

Then again, with the more abstract topics in algebraic geometry, n-categories, stacks and all that stuff, these developments could be carried over to noncommutative geometry, and since NCG is at the heart of many developments in physics, it might give wonderful applications to string theory etc., and in a deeper understanding of our physical world.

Answer 6

There is a lecture course by M. Kontsevich (ENS, 1998) which has a chapter entitled "algebraic geometry from a noncommutative viewpoint". The notes are available here

www.math.uchicago.edu/~mitya/langlands/kontsevich.ps

It mentions in particular Bondal-Orlov's theorem that if a variety is Fano or of general type, then one can reconstruct it from its coherent derived category. There is also a section on Mirror symmetry. Proofs are only sketched, but these sketches can be most illuminating (or sometimes not).

In general, the "noncommutative geometry" section of

http://www.math.uchicago.edu/~mitya/langlands.html

can be useful, but some of the references there (Keller, Lefevre) do not specifically deal with applications to algebraic geometry. (And maybe someone else here can elaborate on those that do.)